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Journal Cover Physica D: Nonlinear Phenomena
  [SJR: 1.049]   [H-I: 102]   [3 followers]  Follow
    
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0167-2789
   Published by Elsevier Homepage  [3039 journals]
  • Rigorous numerics for NLS: Bound states, spectra, and controllability
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Roberto Castelli, Holger Teismann
      In this paper it is demonstrated how rigorous numerics may be applied to the one-dimensional nonlinear Schrödinger equation (NLS); specifically, to determining bound-state solutions and establishing certain spectral properties of the linearization. Since the results are rigorous, they can be used to complete a recent analytical proof (Beauchard et al., 2015) of the local exact controllability of NLS.


      PubDate: 2016-09-13T04:50:37Z
       
  • Automatic differentiation for Fourier series and the radii polynomial
           approach
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Jean-Philippe Lessard, J.D. Mireles James, Julian Ransford
      In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP).


      PubDate: 2016-09-13T04:50:37Z
       
  • On loops in the hyperbolic locus of the complex Hénon map and their
           monodromies
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Zin Arai
      We prove John Hubbard’s conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Hénon map. In fact, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Furthermore, we prove that the dynamics of the real Hénon map is completely determined by the monodromy of the complex Hénon map, providing the parameter of the map is contained in the hyperbolic horseshoe locus.


      PubDate: 2016-09-13T04:50:37Z
       
  • Chaos near a resonant inclination-flip
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Marcus Fontaine, William Kalies, Vincent Naudot
      Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finitely many iterations. In this work we construct a new model by re-injecting the points that escape the horseshoe. We show that this model can be realized within an attractor of a flow arising from a three-dimensional vector field, after perturbation of an inclination-flip homoclinic orbit with a resonance. The dynamics of this model, without considering the re-injection, often contains a cuspidal horseshoe with positive entropy, and we show that for a computational example the dynamics with re-injection can have more complexity than the cuspidal horseshoe alone.


      PubDate: 2016-09-13T04:50:37Z
       
  • Continuation of point clouds via persistence diagrams
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Marcio Gameiro, Yasuaki Hiraoka, Ippei Obayashi
      In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton–Raphson continuation method in this setting. Given an original point cloud P , its persistence diagram D , and a target persistence diagram D ′ , we gradually move from D to D ′ , by successively computing intermediate point clouds until we finally find a point cloud P ′ having D ′ as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.


      PubDate: 2016-09-13T04:50:37Z
       
  • Principal component analysis of persistent homology rank functions with
           case studies of spatial point patterns, sphere packing and colloids
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Vanessa Robins, Katharine Turner
      Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the persistent homology rank functions. For a point pattern X we construct a filtration of spaces by taking the union of balls of radius a centred on points in X , X a = ∪ x ∈ X B ( x , a ) . The rank function β k ( X ) : { ( a , b ) ∈ R 2 : a ≤ b } → R is then defined by β k ( X ) ( a , b ) = rank ( ι ∗ : H k ( X a ) → H k ( X b ) ) where ι ∗ is the induced map on homology from the inclusion map on spaces. We consider the rank functions as lying in a Hilbert space and show that under reasonable conditions the rank functions from multiple simulations or experiments will lie in an affine subspace. This enables us to perform functional principal component analysis which we apply to experimental data from colloids at different effective temperatures and to sphere packings with different volume fractions. We also investigate the potential of rank functions in providing a test of complete spatial randomness of 2D point patterns using the distances to an empirically computed mean rank function of binomial point patterns in the unit square.


      PubDate: 2016-09-13T04:50:37Z
       
  • Analysis of Kolmogorov flow and Rayleigh–Bénard convection
           using persistent homology
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Miroslav Kramár, Rachel Levanger, Jeffrey Tithof, Balachandra Suri, Mu Xu, Mark Paul, Michael F. Schatz, Konstantin Mischaikow
      We use persistent homology to build a quantitative understanding of large complex systems that are driven far-from-equilibrium. In particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh–Bénard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatio-temporal behavior.


      PubDate: 2016-09-13T04:50:37Z
       
  • Exploring the topology of dynamical reconstructions
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Joshua Garland, Elizabeth Bradley, James D. Meiss
      Computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a number of free parameters and the computation of homology for a large number of simplices can be expensive. This paper is a study of how to compute the homology efficiently and effectively without a full (diffeomorphic) reconstruction. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a simplicial complex whose vertices are a small subset of the data: the “witness complex”. Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds for assuring topological conjugacy between the true and reconstructed dynamics that are specified in the embedding theorems. We conjecture that this is because the requirements for reconstructing homology are less stringent: a homeomorphism is sufficient—as opposed to a diffeomorphism, as is necessary for the full dynamics. We provide preliminary evidence that a homeomorphism, in the form of a delay-coordinate reconstruction map, may exist at a lower dimension than that required to achieve an embedding.


      PubDate: 2016-09-13T04:50:37Z
       
  • Topological microstructure analysis using persistence landscapes
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Paweł Dłotko, Thomas Wanner
      Phase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn–Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations.


      PubDate: 2016-09-13T04:50:37Z
       
  • The Poincaré–Bendixson Theorem and the non-linear
           Cauchy–Riemann equations
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): J.B. van den Berg, S. Munaò, R.C.A.M. Vandervorst
      Fiedler and Mallet-Paret (1989) prove a version of the classical Poincaré–Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy–Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.


      PubDate: 2016-09-13T04:50:37Z
       
  • Arnold’s mechanism of diffusion in the spatial circular restricted
           three-body problem: A semi-analytical argument
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Amadeu Delshams, Marian Gidea, Pablo Roldan
      We consider the spatial circular restricted three-body problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3-degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L 1 , located between Sun and Earth. Near L 1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3-sphere. For an orbit confined to this 3-sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly. We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3-sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such ‘diffusing’ orbits, and numerical evidence that the premises of the theorem are satisfied in the three-body problem considered here. We provide an explicit construction of diffusing orbits. The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the ‘outer dynamics’ along homoclinic orbits, and use very little information on the ‘inner dynamics’ restricted to the 3-sphere. As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L 1 from a nearly-planar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.


      PubDate: 2016-09-13T04:50:37Z
       
  • Geometric phase in the Hopf bundle and the stability of non-linear waves
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Colin J. Grudzien, Thomas J. Bridges, Christopher K.R.T. Jones
      We develop a stability index for the traveling waves of non-linear reaction–diffusion equations using the geometric phase induced on the Hopf bundle S 2 n − 1 ⊂ C n . This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeros correspond to the eigenvalues of the linearization of reaction–diffusion operators about the wave. The stability of a traveling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way’s Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on C 2 and sketch the proof of the method of geometric phase for C n and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.


      PubDate: 2016-09-13T04:50:37Z
       
  • Topology in Dynamics, Differential Equations, and Data
    • Abstract: Publication date: 1 November 2016
      Source:Physica D: Nonlinear Phenomena, Volume 334
      Author(s): Sarah Day, Robertus C.A.M. Vandervorst, Thomas Wanner
      This special issue is devoted to showcasing recent uses of topological methods in the study of dynamical behavior and the analysis of both numerical and experimental data. The twelve original research papers span a wide spectrum of results from abstract index theories, over homology- and persistence-based data analysis techniques, to computer-assisted proof techniques based on topological fixed point arguments.


      PubDate: 2016-09-13T04:50:37Z
       
  • Dispersive hydrodynamics: Preface
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): G. Biondini, G.A. El, M.A. Hoefer, P.D. Miller
      This Special Issue on Dispersive Hydrodynamics is dedicated to the memory and work of G.B. Whitham who was one of the pioneers in this field of physical applied mathematics. Some of the papers appearing here are related to work reported on at the workshop “Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications” held in May 2015 at the Banff International Research Station. This Preface provides a broad overview of the field and summaries of the various contributions to the Special Issue, placing them in a unified context.


      PubDate: 2016-09-03T04:24:00Z
       
  • Integrable extended van der Waals model
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Francesco Giglio, Giulio Landolfi, Antonio Moro
      Inspired by the recent developments in the study of the thermodynamics of van der Waals fluids via the theory of nonlinear conservation laws and the description of phase transitions in terms of classical (dissipative) shock waves, we propose a novel approach to the construction of multi-parameter generalisations of the van der Waals model. The theory of integrable nonlinear conservation laws still represents the inspiring framework. Starting from a macroscopic approach, a four parameter family of integrable extended van der Waals models is indeed constructed in such a way that the equation of state is a solution to an integrable nonlinear conservation law linearisable by a Cole–Hopf transformation. This family is further specified by the request that, in regime of high temperature, far from the critical region, the extended model reproduces asymptotically the standard van der Waals equation of state. We provide a detailed comparison of our extended model with two notable empirical models such as Peng–Robinson and Soave’s modification of the Redlich–Kwong equations of state. We show that our extended van der Waals equation of state is compatible with both empirical models for a suitable choice of the free parameters and can be viewed as a master interpolating equation. The present approach also suggests that further generalisations can be obtained by including the class of dispersive and viscous-dispersive nonlinear conservation laws and could lead to a new type of thermodynamic phase transitions associated to nonclassical and dispersive shock waves.


      PubDate: 2016-09-03T04:24:00Z
       
  • Sine–Gordon modulation solutions: Application to macroscopic
           non-lubricant friction
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Naum I. Gershenzon, Gust Bambakidis, Thomas E. Skinner
      The Frenkel–Kontorova (FK) model and its continuum approximation, the sine–Gordon (SG) equation, are widely used to model a variety of important nonlinear physical systems. Many practical applications require the wave-train solution, which includes many solitons. In such cases, an important and relevant extension of these models applies Whitham’s averaging procedure to the SG equation. The resulting SG modulation equations describe the behavior of important measurable system parameters that are the average of the small-scale solutions given by the SG equation. A fundamental problem of modern physics that is the topic of this paper is the description of the transitional process from a static to a dynamic frictional regime. We have shown that the SG modulation equations are a suitable apparatus for describing this transition. The model provides relations between kinematic (rupture and slip velocities) and dynamic (shear and normal stresses) parameters of the transition process. A particular advantage of the model is its ability to describe frictional processes over a wide range of rupture and slip velocities covering seismic events ranging from regular earthquakes, with rupture velocities on the order of a few km/s, to slow slip events, with rupture velocities on the order of a few km/day.


      PubDate: 2016-09-03T04:24:00Z
       
  • Observation of dispersive shock waves developing from initial depressions
           in shallow water
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): S. Trillo, M. Klein, G.F. Clauss, M. Onorato
      We investigate surface gravity waves in a shallow water tank, in the limit of long wavelengths. We report the observation of non-stationary dispersive shock waves rapidly expanding over a 90 m flume. They are excited by means of a wave maker that allows us to launch a controlled smooth (single well) depression with respect to the unperturbed surface of the still water, a case that contains no solitons. The dynamics of the shock waves are observed at different levels of nonlinearity equivalent to a different relative smallness of the dispersive effect. The observed undulatory behavior is found to be in good agreement with the dynamics described in terms of a Korteweg–de Vries equation with evolution in space, though in the most nonlinear cases the description turns out to be improved over the quasi linear trailing edge of the shock by modeling the evolution in terms of the integro-differential (nonlocal) Whitham equation.


      PubDate: 2016-09-03T04:24:00Z
       
  • Interaction of solitons with long waves in a rotating fluid
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): L.A. Ostrovsky, Y.A. Stepanyants
      Interaction of a soliton with long background waves is studied within the framework of rotation modified Korteweg–de Vries (rKdV) equation. Using the asymptotic method for solitons propagating in the field of a long background wave we derive a set of ODEs describing soliton amplitude and phase with respect to the background wave. The shape of the background wave may range from a sinusoid to the limiting profile representing a periodic sequence of parabolic arcs. We analyse energy exchange between a soliton and the long wave taking radiation losses into account. It is shown that the losses can be compensated by energy pumping from the long wave and, as the result, a stationary soliton can exist, unlike the case when there is no variable background. A more complex case when a free long wave attenuates due to the energy consumption by a soliton is also considered. Some of the analytical results are compared with the results of direct numerical calculations within the framework of the rKdV equation.


      PubDate: 2016-09-03T04:24:00Z
       
  • Generation of polarization singularities in the self-focusing of an
           elliptically polarized laser beam in an isotropic Kerr medium
    • Abstract: Publication date: 1 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 332
      Author(s): N.A. Panov, V.A. Makarov, K.S. Grigoriev, M.S. Yatskevitch, O.G. Kosareva
      We have numerically and analytically shown that polarization singularities can emerge when a homogeneously elliptically polarized light beam undergoes self-focusing in an isotropic third-order Kerr medium without frequency and spatial dispersion (fused silica, liquids, gases etc.) In the case of axially symmetric beam the emerging C -lines have the shape of circumference with the center at the beam’s axis and they are located in the separate transversal planes in the medium. If the axial symmetry of the incident beam is broken then the even number of C -points with opposite topological charges are nucleated in the medium. They exist in a certain propagation coordinate range and then they collide and annihilate each other.


      PubDate: 2016-09-03T04:24:00Z
       
  • Nonlinear random optical waves: Integrable turbulence, rogue waves and
           intermittency
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Stéphane Randoux, Pierre Walczak, Miguel Onorato, Pierre Suret
      We examine the general question of statistical changes experienced by ensembles of nonlinear random waves propagating in systems ruled by integrable equations. In our study that enters within the framework of integrable turbulence, we specifically focus on optical fiber systems accurately described by the integrable one-dimensional nonlinear Schrödinger equation. We consider random complex fields having a Gaussian statistics and an infinite extension at initial stage. We use numerical simulations with periodic boundary conditions and optical fiber experiments to investigate spectral and statistical changes experienced by nonlinear waves in focusing and in defocusing propagation regimes. As a result of nonlinear propagation, the power spectrum of the random wave broadens and takes exponential wings both in focusing and in defocusing regimes. Heavy-tailed deviations from Gaussian statistics are observed in focusing regime while low-tailed deviations from Gaussian statistics are observed in defocusing regime. After some transient evolution, the wave system is found to exhibit a statistically stationary state in which neither the probability density function of the wave field nor the spectrum changes with the evolution variable. Separating fluctuations of small scale from fluctuations of large scale both in focusing and defocusing regimes, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian.


      PubDate: 2016-09-03T04:24:00Z
       
  • Traveling waves for a model of gravity-driven film flows in cylindrical
           domains
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Roberto Camassa, Jeremy L. Marzuola, H. Reed Ogrosky, Nathan Vaughn
      Traveling wave solutions are studied for a recently-derived model of a falling viscous film on the interior of a vertical rigid tube. By identifying a Hopf bifurcation and using numerical continuation software, families of non-trivial traveling wave solutions may be traced out in parameter space. These families all contain a single solution at a ‘turnaround point’ with larger film thickness than all others in the family. In an earlier paper, it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model. Here, these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques. The linear stability of these solutions is also discussed; both large- and small-amplitude solutions are shown to be unstable, though the instability mechanisms are different for each wave type. Specifically, for small-amplitude waves, the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat-film solution unstable; for large-amplitude waves, this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small-amplitude waves.


      PubDate: 2016-09-03T04:24:00Z
       
  • Dispersive shock waves and modulation theory
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): G.A. El, M.A. Hoefer
      There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G.B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs.


      PubDate: 2016-09-03T04:24:00Z
       
  • Self-focusing dynamics of patches of ripples
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): P.A. Milewski, Z. Wang
      The dynamics of focussing of extended patches of nonlinear capillary–gravity waves within the primitive fluid dynamic equations is presented. It is found that, when the envelope has certain properties, the patch focusses initially in accordance to predictions from nonlinear Schrödinger equation, and focussing can concentrate energy to the vicinity of a point or a curve on the fluid surface. After initial focussing, other effects dominate and the patch breaks up into a complex set of localised structures–lumps and breathers–plus dispersive radiation. We perform simulations both in the inviscid regime and for small viscosities. Lastly we discuss throughout the similarities and differences between the dynamics of ripple patches and self-focussing light beams.


      PubDate: 2016-09-03T04:24:00Z
       
  • Mechanical balance laws for fully nonlinear and weakly dispersive water
           waves
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Henrik Kalisch, Zahra Khorsand, Dimitrios Mitsotakis
      The Serre–Green–Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is known to describe accurately the wave motion at the surface of an incompressible inviscid fluid in the case when the fluid flow is irrotational and two-dimensional. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre–Green–Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre–Green–Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling. One consequence of the present analysis is that the energy loss appearing in the shallow-water theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre–Green–Naghdi equations using a finite-element discretization coupled with an adaptive Runge–Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre–Green–Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling.


      PubDate: 2016-09-03T04:24:00Z
       
  • Nonlinear disintegration of sine wave in the framework of the Gardner
           equation
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Oxana Kurkina, Ekaterina Rouvinskaya, Tatiana Talipova, Andrey Kurkin, Efim Pelinovsky
      Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg–de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative soliton-like pulses. This is the main difference with respect to the classic Korteweg–de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of two-soliton interaction of “exchange” or “overtake” types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when “free” velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k 4 / 3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.


      PubDate: 2016-09-03T04:24:00Z
       
  • Nonlinear ring waves in a two-layer fluid
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Karima R. Khusnutdinova, Xizheng Zhang
      Surface and interfacial weakly-nonlinear ring waves in a two-layer fluid are modelled numerically, within the framework of the recently derived 2 + 1 -dimensional cKdV-type equation. In a case study, we consider concentric waves from a localised initial condition and waves in a 2D version of the dam-break problem, as well as discussing the effect of a piecewise-constant shear flow. The modelling shows, in particular, the formation of 2D dispersive shock waves and oscillatory wave trains.


      PubDate: 2016-09-03T04:24:00Z
       
  • The scattering transform for the Benjamin–Ono equation in the
           small-dispersion limit
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Peter D. Miller, Alfredo N. Wetzel
      Using exact formulae for the scattering data of the Benjamin–Ono equation valid for general rational potentials recently obtained in Miller and Wetzel [17], we rigorously analyze the scattering data in the small-dispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself. Our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering, and they also provide new details not previously reported in the literature.


      PubDate: 2016-09-03T04:24:00Z
       
  • The propagation of internal undular bores over variable topography
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): R. Grimshaw, C. Yuan
      In the coastal ocean, large amplitude, horizontally propagating internal wave trains are commonly observed. These are long nonlinear waves and can be modelled by equations of the Korteweg–de Vries type. Typically they occur in regions of variable bottom topography when the variable-coefficient Korteweg–de Vries equation is an appropriate model. Of special interest is the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here we examine the same situation for an undular bore, represented by a modulated periodic wave train. Numerical simulations and some asymptotic analysis based on Whitham modulation equations show that the leading solitary waves in the undular bore are destroyed and replaced by a developing rarefaction wave supporting emerging solitary waves of the opposite polarity. In contrast the rear of the undular bore emerges with the same shape, but with reduced wave amplitudes, a shorter overall length scale and moves more slowly.


      PubDate: 2016-09-03T04:24:00Z
       
  • On critical behaviour in generalized Kadomtsev–Petviashvili
           equations
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): B. Dubrovin, T. Grava, C. Klein
      An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev–Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlevé I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the dispersive shock waves.


      PubDate: 2016-09-03T04:24:00Z
       
  • Semiclassical limit of the focusing NLS: Whitham equations and the
           Riemann–Hilbert Problem approach
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Alexander Tovbis, Gennady A. El
      The main goal of this paper is to put together: a) the Whitham theory applicable to slowly modulated N -phase nonlinear wave solutions to the focusing nonlinear Schrödinger (fNLS) equation, and b) the Riemann–Hilbert Problem approach to particular solutions of the fNLS in the semiclassical (small dispersion) limit that develop slowly modulated N -phase nonlinear wave in the process of evolution. Both approaches have their own merits and limitations. Understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fNLS.


      PubDate: 2016-09-03T04:24:00Z
       
  • Primitive potentials and bounded solutions of the KdV equation
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): S. Dyachenko, D. Zakharov, V. Zakharov
      We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.


      PubDate: 2016-09-03T04:24:00Z
       
  • Small dispersion limit of the Korteweg–de Vries equation with periodic
           initial conditions and analytical description of the Zabusky–Kruskal
           experiment
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Guo Deng, Gino Biondini, Stefano Trillo
      We study the small dispersion limit of the Korteweg–de Vries (KdV) equation with periodic boundary conditions and we apply the results to the Zabusky–Kruskal experiment. In particular, we employ a WKB approximation for the solution of the scattering problem for the KdV equation [i.e., the time-independent Schrödinger equation] to obtain an asymptotic expression for the trace of the monodromy matrix and thereby of the spectrum of the problem. We then perform a detailed analysis of the structure of said spectrum (i.e., band widths, gap widths and relative band widths) as a function of the dispersion smallness parameter ϵ . We then formulate explicit approximations for the number of solitons and corresponding soliton amplitudes as a function of ϵ . Finally, by performing an appropriate rescaling, we compare our results to those in the famous Zabusky and Kruskal’s paper, showing very good agreement with the numerical results.


      PubDate: 2016-09-03T04:24:00Z
       
  • Inverse scattering transform for the defocusing nonlinear Schrödinger
           equation with fully asymmetric non-zero boundary conditions
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Gino Biondini, Emily Fagerstrom, Barbara Prinari
      We formulate the inverse scattering transform (IST) for the defocusing nonlinear Schrödinger (NLS) equation with fully asymmetric non-zero boundary conditions (i.e., when the limiting values of the solution at space infinities have different non-zero moduli). The theory is formulated without making use of Riemann surfaces, and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem. For the direct problem, we give explicit single-valued definitions of the Jost eigenfunctions and scattering coefficients over the whole complex plane, and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues. We pose the inverse problem as a Riemann–Hilbert Problem on an open contour, and we reduce the problem to a standard set of linear integral equations. Finally, for comparison purposes, we present the single-sheet, branch cut formulation of the inverse scattering transform for the initial value problem with symmetric (equimodular) non-zero boundary conditions, as well as for the initial value problem with one-sided non-zero boundary conditions, and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts.


      PubDate: 2016-09-03T04:24:00Z
       
  • Whitham modulation equations, coalescing characteristics, and dispersive
           Boussinesq dynamics
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Daniel J. Ratliff, Thomas J. Bridges
      Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1 + 1 and 2 + 1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and do not depend on particular equations. One curious by-product of the analysis is that the theory can be used to confirm that the two-way Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein–Gordon equation is used to illustrate the theory.


      PubDate: 2016-09-03T04:24:00Z
       
  • Whitham theory for perturbed Korteweg–de Vries equation
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): A.M. Kamchatnov
      Original Whitham’s method of derivation of modulation equations is applied to systems whose dynamics is described by a perturbed Korteweg–de Vries equation. Two situations are distinguished: (i) the perturbation leads to appearance of right-hand sides in the modulation equations so that they become non-uniform; (ii) the perturbation leads to modification of the matrix of Whitham velocities. General form of Whitham modulation equations is obtained in both cases. The essential difference between them is illustrated by an example of so-called ‘generalized Korteweg–de Vries equation’. Method of finding steady-state solutions of perturbed Whitham equations in the case of dissipative perturbations is considered.


      PubDate: 2016-09-03T04:24:00Z
       
  • On the generation of dispersive shock waves
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Peter D. Miller
      We review various methods for the analysis of initial-value problems for integrable dispersive equations in the weak-dispersion or semiclassical regime. Some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains, so-called dispersive shock waves, as the result of shock formation in a limiting dispersionless system. They also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves, revealing fascinating universal wave patterns.


      PubDate: 2016-09-03T04:24:00Z
       
  • Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional
           Benjamin–Ono equations
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Mark J. Ablowitz, Ali Demirci, Yi-Ping Ma
      Dispersive shock waves (DSWs) in the Kadomtsev–Petviashvili (KP) equation and two dimensional Benjamin–Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time ( 2 + 1 ) dimensions to finding DSW solutions of ( 1 + 1 ) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg–de Vries (cKdV) and cylindrical Benjamin–Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the ( 2 + 1 ) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced ( 1 + 1 ) dimensional equations.


      PubDate: 2016-09-03T04:24:00Z
       
  • Modulation theory, dispersive shock waves and Gerald Beresford Whitham
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): A.A. Minzoni, Noel F. Smyth
      Gerald Beresford (GB) Whitham, FRS, (13th December, 1927–26th January, 2014) was one of the leading applied mathematicians of the twentieth century whose work over forty years had a profound, formative impact on research on wave motion across a broad range of areas. Many of the ideas and techniques he developed have now become the standard tools used to analyse and understand wave motion, as the papers of this special issue of Physica D testify. Many of the techniques pioneered by GB Whitham have spread beyond wave propagation into other applied mathematics areas, such as reaction–diffusion, and even into theoretical physics and pure mathematics, in which Whitham modulation theory is an active area of research. GB Whitham’s classic textbook Linear and Nonlinear Waves, published in 1974, is still the standard reference for the applied mathematics of wave motion. In honour of his scientific achievements, GB Whitham was elected a Fellow of the American Academy of Arts and Sciences in 1959 and a Fellow of the Royal Society in 1965. He was awarded the Norbert Wiener Prize for Applied Mathematics in 1980.


      PubDate: 2016-09-03T04:24:00Z
       
  • Incoherent shock waves in long-range optical turbulence
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): G. Xu, J. Garnier, D. Faccio, S. Trillo, A. Picozzi
      Considering the nonlinear Schrödinger (NLS) equation as a representative model, we report a unified presentation of different forms of incoherent shock waves that emerge in the long-range interaction regime of a turbulent optical wave system. These incoherent singularities can develop either in the temporal domain through a highly noninstantaneous nonlinear response, or in the spatial domain through a highly nonlocal nonlinearity. In the temporal domain, genuine dispersive shock waves (DSW) develop in the spectral dynamics of the random waves, despite the fact that the causality condition inherent to the response function breaks the Hamiltonian structure of the NLS equation. Such spectral incoherent DSWs are described in detail by a family of singular integro-differential kinetic equations, e.g. Benjamin–Ono equation, which are derived from a nonequilibrium kinetic formulation based on the weak Langmuir turbulence equation. In the spatial domain, the system is shown to exhibit a large scale global collective behavior, so that it is the fluctuating field as a whole that develops a singularity, which is inherently an incoherent object made of random waves. Despite the Hamiltonian structure of the NLS equation, the regularization of such a collective incoherent shock does not require the formation of a DSW — the regularization is shown to occur by means of a different process of coherence degradation at the shock point. We show that the collective incoherent shock is responsible for an original mechanism of spontaneous nucleation of a phase-space hole in the spectrogram dynamics. The robustness of such a phase-space hole is interpreted in the light of incoherent dark soliton states, whose different exact solutions are derived in the framework of the long-range Vlasov formalism.


      PubDate: 2016-09-03T04:24:00Z
       
  • Dispersive shock waves in nematic liquid crystals
    • Abstract: Publication date: 15 October 2016
      Source:Physica D: Nonlinear Phenomena, Volume 333
      Author(s): Noel F. Smyth
      The propagation of coherent light with an initial step intensity profile in a nematic liquid crystal is studied using modulation theory. The propagation of light in a nematic liquid crystal is governed by a coupled system consisting of a nonlinear Schrödinger equation for the light beam and an elliptic equation for the medium response. In general, the intensity step breaks up into a dispersive shock wave, or undular bore, and an expansion fan. In the experimental parameter regime for which the nematic response is highly nonlocal, this nematic bore is found to differ substantially from the standard defocusing nonlinear Schrödinger equation structure due to the effect of the nonlocality of the nematic medium. It is found that the undular bore is of Korteweg–de Vries equation-type, consisting of bright waves, rather than of nonlinear Schrödinger equation-type, consisting of dark waves. In addition, ahead of this Korteweg–de Vries bore there can be a uniform wavetrain with a short front which brings the solution down to the initial level ahead. It is found that this uniform wavetrain does not exist if the initial jump is below a critical value. Analytical solutions for the various parts of the nematic bore are found, with emphasis on the role of the nonlocality of the nematic medium in shaping this structure. Excellent agreement between full numerical solutions of the governing nematicon equations and these analytical solutions is found.


      PubDate: 2016-09-03T04:24:00Z
       
  • Fractional Schrödinger dynamics and decoherence
    • Abstract: Publication date: Available online 16 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Kay Kirkpatrick, Yanzhi Zhang
      We study the dynamics of the Schrödinger equation with a fractional Laplacian ( − Δ ) α , and the decoherence of the solution is observed. Analytically, we obtain equations of motion for the expected position and momentum in the fractional Schödinger equation, equations that are the fractional counterpart of the well-known Newtonian equations of motion for the standard ( α = 1 ) Schrödinger equation. Numerically, we propose an explicit, effective numerical method for solving the time-dependent fractional nonlinear Schrödinger equation–a method that has high order spatial accuracy, requires little memory, and has low computational cost. We apply our method to study the dynamics of fractional Schrödinger equation and find that the nonlocal interactions from the fractional Laplacian introduce decoherence into the solution. The local nonlinear interactions can however reduce or delay the emergence of decoherence. Moreover, we find that the solution of the standard NLS behaves more like a particle, but the solution of the fractional NLS behaves more like a wave with interference effects.


      PubDate: 2016-06-18T18:04:02Z
       
  • Entropy rates of low-significance bits sampled from chaotic physical
           systems
    • Abstract: Publication date: Available online 16 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Ned J. Corron, Roy M. Cooper, Jonathan N. Blakely
      We examine the entropy of low-significance bits in analog-to-digital measurements of chaotic dynamical systems. We find the partition of measurement space corresponding to low-significance bits has a corrugated structure. Using simulated measurements of a map and experimental data from a circuit, we identify two consequences of this corrugated partition. First, entropy rates for sequences of low-significance bits more closely approach the metric entropy of the chaotic system, because the corrugated partition better approximates a generating partition. Second, accurate estimation of the entropy rate using low-significance bits requires long block lengths as the corrugated partition introduces more long-term correlation, and using only short block lengths overestimates the entropy rate. This second phenomenon may explain recent reports of experimental systems producing binary sequences that pass statistical tests of randomness at rates that may be significantly beyond the metric entropy rate of the physical source.


      PubDate: 2016-06-18T18:04:02Z
       
  • Nonlinear wave dynamics near phase transition in PT-symmetric localized
           potentials
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): Sean Nixon, Jianke Yang
      Nonlinear wave propagation in parity-time symmetric localized potentials is investigated analytically near a phase-transition point where a pair of real eigenvalues of the potential coalesce and bifurcate into the complex plane. Necessary conditions for a phase transition to occur are derived based on a generalization of the Krein signature. Using the multi-scale perturbation analysis, a reduced nonlinear ordinary differential equation (ODE) is derived for the amplitude of localized solutions near phase transition. Above the phase transition, this ODE predicts a family of stable solitons not bifurcating from linear (infinitesimal) modes under a certain sign of nonlinearity. In addition, it predicts periodically-oscillating nonlinear modes away from solitons. Under the opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below the phase transition, solution dynamics is predicted as well. All analytical results are compared to direct computations of the full system and good agreement is observed.


      PubDate: 2016-06-18T18:04:02Z
       
  • Degenerate Bogdanov–Takens bifurcations in a one-dimensional transport
           model of a fusion plasma
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): H.J. de Blank, Yu.A. Kuznetsov, M.J. Pekkér, D.W.M. Veldman
      Experiments in tokamaks (nuclear fusion reactors) have shown two modes of operation: L-mode and H-mode. Transitions between these two modes have been observed in three types: sharp, smooth and oscillatory. The same modes of operation and transitions between them have been observed in simplified transport models of the fusion plasma in one spatial dimension. We study the dynamics in such a one-dimensional transport model by numerical continuation techniques. To this end the MATLAB package cl_matcontL was extended with the continuation of (codimension-2) Bogdanov–Takens bifurcations in three parameters using subspace reduction techniques. During the continuation of (codimension-2) Bogdanov–Takens bifurcations in 3 parameters, generically degenerate Bogdanov–Takens bifurcations of codimension-3 are detected. However, when these techniques are applied to the transport model, we detect a degenerate Bogdanov–Takens bifurcation of codimension 4. The nearby 1- and 2-parameter slices are in agreement with the presence of this codimension-4 degenerate Bogdanov–Takens bifurcation, and all three types of L–H transitions can be recognized in these slices. The same codimension-4 situation is observed under variation of the additional parameters in the model, and under some modifications of the model.


      PubDate: 2016-06-18T18:04:02Z
       
  • Burgers equation with no-flux boundary conditions and its application for
           complete fluid separation
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): Shinya Watanabe, Sohei Matsumoto, Tomohiro Higurashi, Naoki Ono
      Burgers equation in a one-dimensional bounded domain with no-flux boundary conditions at both ends is proven to be exactly solvable. Cole–Hopf transformation converts not only the governing equation to the heat equation with an extra damping but also the nonlinear mixed boundary conditions to Dirichlet boundary conditions. The average of the solution v ¯ is conserved. Consequently, from an arbitrary initial condition, solutions converge to the equilibrium solution which is unique for the given v ¯ . The problem arises naturally as a continuum limit of a network of certain micro-devices. Each micro-device imperfectly separates a target fluid component from a mixture of more than one component, and its input–output concentration relationships are modeled by a pair of quadratic maps. The solvability of the initial boundary value problem is used to demonstrate that such a network acts as an ideal macro-separator, separating out the target component almost completely. Another network is also proposed which leads to a modified Burgers equation with a nonlinear diffusion coefficient.


      PubDate: 2016-06-15T08:40:59Z
       
  • Combustion waves in hydraulically resistant porous media in a special
           parameter regime
    • Abstract: Publication date: Available online 7 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Anna Ghazaryan, Stéphane Lafortune, Peter McLarnan
      In this paper we study the stability of fronts in a reduction of a well-known PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level.


      PubDate: 2016-06-15T08:40:59Z
       
  • Mixed-Mode Oscillations in a piecewise linear system with multiple time
           scale coupling
    • Abstract: Publication date: Available online 10 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Fernández-García, M. Krupa, F. Clément
      In this work, we analyze a four dimensional slow-fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse.


      PubDate: 2016-06-15T08:40:59Z
       
  • A principle of similarity for nonlinear vibration absorbers
    • Abstract: Publication date: Available online 11 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): G. Habib, G. Kerschen
      This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A compact analytical formula for the nonlinear coefficient of the absorber is then derived. The formula, valid for any polynomial nonlinearity in the primary system, is found to depend only on the mass ratio and on the nonlinear coefficient of the primary system. When the primary system comprises several polynomial nonlinearities, we demonstrate that the NLTVA obeys a principle of additivity, i.e., each nonlinear coefficient can be calculated independently of the other nonlinear coefficients using the proposed formula.


      PubDate: 2016-06-15T08:40:59Z
       
  • Resonance Van Hove singularities in wave kinetics
    • Abstract: Publication date: Available online 14 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yi-Kang Shi, Gregory L. Eyink
      Wave kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. However, we show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electron-hole excitations (matter waves) in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space D = ( N − 2 ) d ( d physical space dimension, N the number of waves in resonance) and the degree of degeneracy δ of the critical points. Following Van Hove, we show that non-degenerate singularities lead to finite phase measures for D > 2 but produce divergences when D ≤ 2 and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when D − δ ≤ 2 , as we find for several physical examples, including electron-hole kinetics in graphene. When the standard kinetic equation breaks down, then one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell & Aucoin on multi-scale perturbation theory for acoustic waves and field-theoretic methods based on exact Schwinger-Dyson integral equations for the wave dynamics.


      PubDate: 2016-06-15T08:40:59Z
       
  • Breathers in a locally resonant granular chain with precompression
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): Lifeng Liu, Guillaume James, Panayotis Kevrekidis, Anna Vainchtein
      We study a locally resonant granular material in the form of a precompressed Hertzian chain with linear internal resonators. Using an asymptotic reduction, we derive an effective nonlinear Schrödinger (NLS) modulation equation. This, in turn, leads us to provide analytical evidence, subsequently corroborated numerically, for the existence of two distinct types of discrete breathers related to acoustic or optical modes: (a) traveling bright breathers with a strain profile exponentially vanishing at infinity and (b) stationary and traveling dark breathers, exponentially localized, time-periodic states mounted on top of a non-vanishing background. The stability and bifurcation structure of numerically computed exact stationary dark breathers is also examined. Stationary bright breathers cannot be identified using the NLS equation, which is defocusing at the upper edges of the phonon bands and becomes linear at the lower edge of the optical band.


      PubDate: 2016-06-15T08:40:59Z
       
 
 
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